1 pound fingerling potatoes, halved lengthwise
2 tablespoons extra-virgin olive oil
2 tablespoons plus 2 teaspoons herbes de Provence, divided
Kosher salt and freshly ground black pepper
1/4 cup unsalted butter, softened
1 tablespoon Dijon mustard
2 (12-ounce) butterflied branzino or trout
Preheat oven to 375°F. Line a rimmed baking sheet with aluminum foil.
Toss together potatoes, olive oil, 2 tablespoons herbes de Provence, 1 teaspoon salt, and 1/2 teaspoon pepper on prepared baking sheet. Roast in preheated oven until potatoes are golden around edges and nearly tender, about 20 minutes.
Meanwhile, mash butter, mustard, remaining 2 teaspoons herbes de Provence, and 1 teaspoon salt together with a fork in a small bowl. Arrange fish flesh side up, season with salt and pepper, and spread butter mixture evenly on fish.
Flip potatoes and arrange around edges of pan to accommodate fish. Add fish to baking sheet, flesh side up, and roast at 375°F until fish is opaque and flaky and potatoes are cooked through, about 10 minutes.